Statistical Methods in Financial Markets
An important complement to differential equation analysis is statistical time series methodology. The advantage of time series is that the model is chosen in a prescribed deductive manner which provides further evidence of the fundamental forces that govern price dynamics beyond valuation. In particular the Box-Jenkins methodology for selecting the appropriate ARIMA model. This procedure then selects from a large collection of models that include a simple random walk, a pure momentum model and more complicated models.
Thus, an ARIMA analysis of a data set has the potential to determine, for example, whether today price derivative predicts tomorrow's price derivative. Acquiring an understanding of markets is greatly complicated by ''noise'' or random events that affect valuation, so that any deterministic forces are difficult to detect.
One way to extract the deterministic forces from the random events is to utilize ratios of pairs of stock that are identical in fundamental value. An ARIMA model extracted from this ratio data can be compared to others and to data compiled from the laboratory experiments. In particular this provides a quantitative link that is very convincing.
The ARIMA methodology can also be utilized in deriving the terms of the differential equations in an empirical way.